Animating a plane in the graph editor with local transform orientation
You cannot manipulate keyframes in local transform, they are stored and displayed in global transform only.

But you can parent the plane to an Empty and orient the plane through it. The plane's global animation is then transformed into local space of the Empty. This might be useful for simple short movements, but non-usable for larger animations. With more complicated movement you could have multiple empties and a dynamic child-of constraint to influence the plane only on sections of the animation. How to setup that is covered here:

How can I build a versatile camera rig which can both orbit around its focus and pan?